Optimal. Leaf size=360 \[ -\frac{\left (21 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (21 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (5 \sqrt{a} e^2+21 \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (5 \sqrt{a} e^2+21 \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{3 d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.326419, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.588, Rules used = {1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (21 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (21 \sqrt{c} d^2-5 \sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (5 \sqrt{a} e^2+21 \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (5 \sqrt{a} e^2+21 \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{3 d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1855
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+c x^4\right )^3} \, dx &=\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2}-\frac{\int \frac{-7 d^2-12 d e x-5 e^2 x^2}{\left (a+c x^4\right )^2} \, dx}{8 a}\\ &=\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{\int \frac{21 d^2+24 d e x+5 e^2 x^2}{a+c x^4} \, dx}{32 a^2}\\ &=\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{\int \left (\frac{24 d e x}{a+c x^4}+\frac{21 d^2+5 e^2 x^2}{a+c x^4}\right ) \, dx}{32 a^2}\\ &=\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{\int \frac{21 d^2+5 e^2 x^2}{a+c x^4} \, dx}{32 a^2}+\frac{(3 d e) \int \frac{x}{a+c x^4} \, dx}{4 a^2}\\ &=\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{(3 d e) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{8 a^2}+\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{64 a^2 c}+\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}+5 e^2\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{64 a^2 c}\\ &=\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{3 d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}-\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} a^{9/4} c^{3/4}}+\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}+5 e^2\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 c}+\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}+5 e^2\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^2 c}\\ &=\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{3 d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}-\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}+\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}+\frac{\left (21 \sqrt{c} d^2+5 \sqrt{a} e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (21 \sqrt{c} d^2+5 \sqrt{a} e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}\\ &=\frac{x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac{x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac{3 d e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}-\frac{\left (21 \sqrt{c} d^2+5 \sqrt{a} e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (21 \sqrt{c} d^2+5 \sqrt{a} e^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}+\frac{\left (\frac{21 \sqrt{c} d^2}{\sqrt{a}}-5 e^2\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.298442, size = 358, normalized size = 0.99 \[ \frac{\frac{\sqrt{2} \left (5 a^{3/4} e^2-21 \sqrt [4]{a} \sqrt{c} d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}+\frac{\sqrt{2} \left (21 \sqrt [4]{a} \sqrt{c} d^2-5 a^{3/4} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}+\frac{32 a^2 x (d+e x)^2}{\left (a+c x^4\right )^2}-\frac{2 \sqrt [4]{a} \left (48 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt{2} \sqrt{a} e^2+21 \sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{2 \sqrt [4]{a} \left (-48 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt{2} \sqrt{a} e^2+21 \sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{8 a x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{a+c x^4}}{256 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 419, normalized size = 1.2 \begin{align*}{\frac{{d}^{2}x}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{7\,{d}^{2}x}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{21\,{d}^{2}\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{21\,{d}^{2}\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{21\,{d}^{2}\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{de{x}^{2}}{4\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{3\,de{x}^{2}}{8\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{3\,de}{8\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{2}{x}^{3}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{5\,{e}^{2}{x}^{3}}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{5\,{e}^{2}\sqrt{2}}{256\,{a}^{2}c}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,{e}^{2}\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,{e}^{2}\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.51826, size = 374, normalized size = 1.04 \begin{align*} \operatorname{RootSum}{\left (268435456 t^{4} a^{11} c^{3} + 25755648 t^{2} a^{6} c^{2} d^{2} e^{2} + t \left (307200 a^{4} c d e^{5} - 5419008 a^{3} c^{2} d^{5} e\right ) + 625 a^{2} e^{8} + 111906 a c d^{4} e^{4} + 194481 c^{2} d^{8}, \left ( t \mapsto t \log{\left (x + \frac{262144000 t^{3} a^{10} c^{2} e^{6} + 46110081024 t^{3} a^{9} c^{3} d^{4} e^{2} - 1645608960 t^{2} a^{7} c^{2} d^{3} e^{5} + 3641573376 t^{2} a^{6} c^{3} d^{7} e + 32688000 t a^{5} c d^{2} e^{8} + 3128219136 t a^{4} c^{2} d^{6} e^{4} + 522764928 t a^{3} c^{3} d^{10} + 225000 a^{3} d e^{11} - 43338240 a^{2} c d^{5} e^{7} - 523431720 a c^{2} d^{9} e^{3}}{15625 a^{3} e^{12} - 21357225 a^{2} c d^{4} e^{8} - 376741449 a c^{2} d^{8} e^{4} + 85766121 c^{3} d^{12}} \right )} \right )\right )} + \frac{11 a d^{2} x + 20 a d e x^{2} + 9 a e^{2} x^{3} + 7 c d^{2} x^{5} + 12 c d e x^{6} + 5 c e^{2} x^{7}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26103, size = 481, normalized size = 1.34 \begin{align*} \frac{5 \, c x^{7} e^{2} + 12 \, c d x^{6} e + 7 \, c d^{2} x^{5} + 9 \, a x^{3} e^{2} + 20 \, a d x^{2} e + 11 \, a d^{2} x}{32 \,{\left (c x^{4} + a\right )}^{2} a^{2}} + \frac{\sqrt{2}{\left (24 \, \sqrt{2} \sqrt{a c} c^{2} d e + 21 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac{\sqrt{2}{\left (24 \, \sqrt{2} \sqrt{a c} c^{2} d e + 21 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} e^{2}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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